After his graduation from Oxford in 1580, Harriot was first hired by Sir Walter Raleigh as a mathematics tutor; he used his knowledge of astronomy/astrology to provide navigational expertise, help design Raleigh's ships, and serve as his accountant. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation.[4] In addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised a phonetic alphabet to transcribe their Carolina Algonquian language.

Harriot and Manteo spent many days in one another's company; Harriot interrogated Manteo closely about life in the New World and learned much that was to the advantage of the English settlers.[5] In addition, he recorded the sense of awe with which the Native Americans viewed European technology:

"Many things they sawe with us...as mathematical instruments, sea compasses...[and] spring clocks that seemed to goe of themselves - and many other things we had - were so strange unto them, and so farre exceeded their capacities to comprehend the reason and meanes how they should be made and done, that they thought they were rather the works of gods than men."[5]

He made only one expedition, around 1585-86, and spent some time in the New World visiting Roanoke Island off the coast of North Carolina, expanding his knowledge by improving his understanding of the Carolina Algonquian language. As the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition.[6]

His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588 (probably written a year before). The True Report contains an early account of the Native American population encountered by the expedition; it proved very influential upon later English explorers and colonists. He wrote: "Whereby it may be hoped, if means of good government be used, that they may in short time be brought to civility and the embracing of true religion." At the same time, his views of Native Americans' industry and capacity to learn were later largely ignored in favour of the parts of the "True Report" about extractable minerals and resources.

As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a striking resemblance to atomism and modern atomic theory, which he was later accused of believing. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, later influenced Kepler's conjecture.

Harriot's sponsors began to fall from favour: Raleigh was the first, and Harriot's other patron Henry Percy, the Earl of Northumberland, was imprisoned in 1605 in connection with the Gunpowder Plot as he was tclosely connected to one of the conspirators, Thomas Percy.

Harriot himself was interrogated and briefly imprisoned but was soon released. Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberland's mansion as they worked for him and assisted in the teaching of the family's children.[3]

Halley's Comet in 1607 turned Harriot's attention towards astronomy. In early 1609 he bought a "Dutch trunke" (telescope), invented in 1608, and his observations were amongst the first uses of a telescope for astronomy. Harriot is now credited as the first astronomer to draw an astronomical object after viewing it through a telescope: he drew a map of the Moon on 26 July 1609, preceding Galileo by several months.[7][8][9][10] He also observed sunspots in December 1610.[11]

In 1615 or 1616, Harriot wrote to an unknown friend with medical expertise, describing what would have been the reason for the eruption of a cancerous ulcer on his lip. This progressed until 1621, when he was living with a friend named Thomas Buckner on Threadneedle Street, where he died. Sources cited below are among several that describe his condition as a cancer of the nose. In either case, Harriot apparently died from skin cancer.

He died on 2 July 1621, three days after writing his will (discovered by Henry Stevens).[12] His executors posthumously published his Artis Analyticae Praxis on algebra in 1631; Nathaniel Torporley was the intended executor of Harriot's wishes, but Walter Warner in the end pulled the book into shape.[13] It may be a compendium of some of his works but does not represent all that he left unpublished (more than 400 sheets of annotated writing). It isn't directed in a way that follows the manuscripts and it fails to give the full significance of Harriot's writings.[3]

He also studied optics and refraction, and apparently discovered Snell's law 20 years before Snellius did, although it was previously discovered by Ibn Sahl; like so many of his works, this remained unpublished. In Virginia he learned the local Algonquian language, which may have had some effect on his mathematical thinking.[citation needed] He founded the "English school" of algebra. He is also credited with discovering Girard's theorem, although the formula bears Girard's name as he was the first to publish it.[14]

His algebra book Artis Analyticae Praxis[15] (1631) was published posthumously in Latin. Unfortunately the editors did not understand much of his reasoning and removed the parts they did not comprehend such as the negative and complex roots of equations. Because of the dispersion of Harriot's writings the full annotated English translation of the Praxis was not completed until 2007.[16]

The first biography of Harriot was written in 1876 by Henry Stevens of Vermont but not published until 1900[12] fourteen years after his death. The publication was limited to 167 copies and so the work was not widely known until 1972 when a reprint edition appeared.[17] Interest in Harriot revived with the convening of a symposium at the University of Delaware in April, 1971 with the proceedings published by the Oxford University Press in 1974.[18] John W. Shirley the editor (1908-1988) went on to publish A Sourcebook for the Study of Thomas Harriot (1981)[19] and his Harriot biography (1983).[20] The papers of John Shirley have been deposited in the University of Delaware Library.[21]

Harriot's accomplishments remain relatively obscure because he did not publish any of his results and also because many of his manuscripts have been lost; those that survive are sheltered in the British Museum and in the archives of the Percy family at Petworth House (Sussex) and Alnwick Castle (Northumberland).

The Thomas Harriot Plaque in the grounds of Syon House (W. London).

An event was held at Syon House, West London, to celebrate the 400th anniversary of Harriot's first observations of the moon on 26 July 2009. This event, Telescope400,[22] included the unveiling of a plaque to commemorate Harriot by Lord Egremont. The plaque can now be seen by visitors to Syon House, the location of Harriot's historic observations. His drawing made 400 years earlier is believed to be based on the first ever observations of the moon through a telescope. The event (sponsored by the Royal Astronomical Society) was run as part of the International Year of Astronomy (IYA).

The original documents showing Harriot's moon map of c. 1611, observations of Jupiter's satellites, and first observations of sunspots were on display at the Science Museum, London, from 23 July 2009 until the end of IYA.[23]

^By 1613 Harriot had created two maps of the whole moon, with many identifiable features such as lunar craters depicted in their correct relative positions that were not to be improved upon for several decades.